Intra-…rm Bargaining and Matching Frictions in
a Multiple Equilibria Model
Julie Beugnot and Mabel Tidbally
…rst version December 17, 2008; this version April 2, 2009
Work in progresszx
Abstract
In this paper, we combine a monopolistic model à la Dixit and Stiglitz
and a matching model derived from Pissarides (2000) in the case of large
…rms. As in Cahuc and Wasmer (2001), we consider an intra-…rm bargaining model for the wage determination. Moreover, we allow for increasing
returns to scale in the production function leading to multiple equilibria. Finally, based on numerical simulations, we investigate the e¤ects of
stronger competition in the product market, higher unemployment bene…ts and stronger bargaining power of workers on the labor market performances in the case of unique equilibrium and multiple equilibria.
Keywords: matching frictions, monopolistic competition, intra-…rm
bargaining, multiple equilibria
JEL Classi…cation: D43, E24, J41
Corresponding author: LAMETA, University of Montpellier 1, O¢ ce C525, Av. de la
mer - Site Richter, C.S 79606, 34960 Montpellier Cedex 2, France. Tel:+33 (0)4 67 15 83 34;
Fax: +33 (0)4 67 15 84 67. Email address: [email protected]
y INRA LAMETA, 2 Place Viala, 34060 Montpellier Cedex 1, France. Email address:
[email protected]
z This version can be subject to substantial modi…cations. Please don’t quote this document
without authors’permission.
x We would like to thank Pierre Cahuc for his helpful comments and suggestions on this
work.
1
Introduction
The understanding of labor market and its reactions facing economic policies is
a widespread subject in economic literature. Since the 1960’s, the sharp variations of unemployment rates in European countries have induced economists
to consider the possibility of multiple unemployment equilibria. Today, this
view is widely accepted. However, the possibility of multiple equilibria involves
reconsideration in policy design. Thus, papers on economic policies design of
labor market in the case of multiple equilibria have emerged.
Actually, as Cooper (1999) explains it, the existence of multiple equilibria
involves the possibility of di¤erentiated policy implications according to the
equilibrium achieved by the economy as well as the possibility by an appropriate
economic policy to coordinate the economy on the Pareto superior equilibrium
in the case of coordination failure1 .
The search and matching models of Pissarides (2000) have permitted to take
into account the fact that the employment cannot be increased without time and
cost. The …rms must post vacancies, what are costly, and commit on wage with
potential workers, what are time-consuming, before that any match is doing
and production takes place. In his book, Pissarides develops essentially the
case of one worker-one …rm which is more convenient and whose results can be
generalized to a multi-workers …rm. However, assuming a standard bargaining
process between the …rm and the worker, its approach has the main drawback
to exclude the strategic interactions within the …rm.
Stole and Zwiebel’s (1996) paper studies such interactions. Indeed, they
provide an intra…rm bargaining model in which contracts cannot commit the
…rms and its workers to wages and employment. The central assumptions of
this wage setting are to consider that the wage is renegotiated with all workers
after a hiring or laid o¤ and so to treat each worker as a marginal worker. This
bargaining process takes place within a multi-workers …rm and, in this way it
…lls the lack of analysis about the e¤ects of an additional worker wage negotiation on this of incumbents in the Pissarides’ model. As Cahuc and Wasmer
(2004) proved, these two approaches are complementary due to theirs analyzes
of di¤erent mechanisms at work in the labor market. Whereas the Pissarides
model is explicitly dynamic and analyzes the labor market equilibrium in the
presence of search and matching frictions but without strategic interactions
within the …rm, the Stole and Zwiebel’s approach analyzes these latter but in
a static framework. Consequently, a model including both approaches will be
more adequate to understand the labor markets working.
Furthermore, the size of competition degree on the product market has been
also advanced in the understanding of labor markets performances. Blanchard
and Giovazzi (2003) show how the product market deregulation, involving lower
entry cost and higher competition degree, leads to higher real wages and lower
unemployment. Nonetheless, they assume a standard Nash bargaining process.
1 In the case where multiple pareto ranked equilibria exist, there exists coordination failure
if the economy reaches a pareto inferior equilibrium (Cooper, 1999)
2
Ebell and Haefke (2003) analyse similar issue but consider an intra…rm bargaining and show quantitatively that in such a framework the impact of product
market competition on equilibrium unemployment is surprisingly weak.
In this paper, we combine the monopolistic model of Dixit and Stiglitz (1977)
and a matching model derived from this of Pissarides (2000) with intra…rm bargaining and large …rms. In this way, we use a similar framework of this developed
by Ebell and Haefke (2003). Our main contribution is to investigate the case of
increasing returns to scale in the production technology in such a framework.
Indeed, whereas all papers listed above consider either decreasing returns to
scale or constant returns to scale in production technology and so the case of
unique equilibrium; our paper follows the Mortensen (1999)’s paper and considers the case of increasing returns involving the existence of multiple equilibria2 .
Then, we show that in the case of multiple equilibria most of comparative statics
results are weaker than those of unique equilibrium case.
The paper is organized as follows. In section 1, we develop the mathematical
model. After explaining the di¤erent assumptions on …rms, labor and matching frictions, we analyze the …rms’behavior and the wage determination under
intra…rm bargaining mechanism. Then, we deduce the general equilibrium at
the steady state. In section 2, the possible equilibrium cases according to the
size of returns to scale are studied. When returns to scale of production technology are either decreasing or constant, there is a unique equilibrium; whereas
there are either multiple equilibria or none in presence of increasing returns to
scale. Numerical simulations are run in section 3. These simulations allow us to
investigate the e¤ects of a higher degree of competition, higher unemployment
bene…ts and stronger bargaining power of workers. In section 4, we manage a
numeric stability analysis and discuss the previous results.
1
1.1
1.1.1
The model
Hypothesis
Economy
We consider a continuous time model of an economy made up agents which have
the same discounted rate r. The output is produced by multi-workers …rms (or
large …rms). The labor is supplied by workers and each worker supplies one unit
of labor. The labor is the only production factor used in this economy. The
existence of matching frictions implies that …rms need time and ressources to
hire workers.
In order to study the dynamic features of our economy, we solve the model
both in and out of the steady state. In this way, we obtain the global dynamics
of the model (Mortensen, 1999).
2 Multiple equilibria can be also induced by increasing returns in the matching function,
transactions costs, menu costs and others. See Cooper (1999) for a detailed report on this
subject.
3
1.1.2
Firms
We assume a monopolistic competition on the product market. We have a
continuum of identical …rms uniformally distributed on the interval [0; 1]. Each
…rm produces an imperfectly substituable good with others. The …rm i has the
following production function yi = Ani , with > 0 and where yi represent its
output, ni its labor employment and A a technology parameter. The demand
for the …rm i’s output is given by yid = Y (pi =P ) ; with pi the price of the
…rm i, P the general price level, Y the aggregate output and > 1 the demand
elasticity for the good supplied by the …rm i3 .
1.1.3
Labor, matching frictions and negotiated wage
Labor is supplied by a continuum of in…nitely lived and identical workers of size
normalized to one. A worker can be employed or unemployed: All unemployed
workers are assumed to have the same search e¤ort which is normalized to one.
The employment cannot be increased instantaneously. The …rms must post
vacancies in order to recruit. This process incurs a real cost c per unit of time
and per unit of vacancy. Furthermore, we assume that the …rm can post as many
vacancies as necessary and without delay, so vacancies are "jump" variables4 .
Vacancies are matched to the pool of unemployed workers according to the
matching technology: m(u; v) = u v 1 ; where 2 [0; 1] represents the matching elasticity, v the mass of vacancies and u the unemployment rate. This
matching function is assumed to be increasing and concave in each argument,
and homogenous of degree one.
v
Let the labor market tightness = , the probability to …ll a vacant job
u
m(u; v)
per unit of time q( ) =
=
, with q 0 ( ) < 0 and q(0) = +1,
v
and the probability for an unemployed worker to …nd a job per unit of time
d [ q( )]
m(u; v)
= 1 with
> 0. We note that is exogenous to the
q( ) =
u
d
…rms’decision.
At each unit of time, a rate s of existing jobs is destructed. This rate of job
destruction is exogenous in our model. Thus, at the …rm level, the employment
evolves following the law of motion: ni = q( )vi sni : Indeed, at each time,
the employment of the …rm i increases with the vacancies which are …lled and
decreases with the existing jobs which are destructed.
The real wage wi (ni ) is continuously and instantaneously negotiated soon
after new information arrives and is a function of …rm’s employment. Consequently, real wages are also assumed to be jump variables. Although the wage
is individually negotiated, it’s assumed to be the same for all workers in the
…rm due to the homogenous labor assumption (approach of Stole and Zwiebel,
1996).
3 See
appendix A for detailed calculations on the determination of demand function.
assumption makes praticable the investigation of dynamics out-of-steady state here-
4 This
after.
4
1.1.4
Sequence of events
The time schedule of the model can be illustrated by the following diagram:
dt
Firm posts vacancies
Individual wage
bargaining
t
Employment
Diagram 1: Sequence of events
For each unit of time, the sequence of events can be described by this story.
At the beginning of the period, the …rm posts as many vacancies as necessary
to hire in expectation the desired number of workers. It takes its decision while
considering the wage given by the incoming wage bargaining and expecting
the fact that its employment level will have an e¤ect on this. Then, once the
employment level determined, the individual bargaining takes place. The real
wages are negotiated between the …rm and each worker (individual bargaining)
even its incumbents. Indeed, when a worker is hired or laid o¤, the wage is
renegotiated with all workers, so each worker is treated as a marginal worker5 .
1.2
Firm’s behavior
The …rm i maximizes the discounted value of future real pro…ts i ; its state
variable is its current employment of workers ni and its control variable is its
number of posted vacancies vi :
The …rm i open as many vacancies as necessary and without delay to have
the desired employment level leading to the maximization of the discounted
value of future real pro…ts. Its discount rate is r.
5 See
Stole and Zwiebel (1996) for more details about the timing of bargaining session.
5
Its problem is to solve6 :
R +1
= M ax 0 e
V (ni0 )
s:t
rt
vi
i
dt
pi
(yi )yi (ni ) wi (ni )ni cvi
P
ni = q( )vi sni ; ni > 0; ni (0) = ni0
yi (ni ) = Ani
pi d
(y ) = (yid =Y ) 1= ; yid = yi (ni )
P i
wi (ni )
given
=
i
(1)
(2)
(3)
(4)
(5)
We assume that the …rm produces exactly the demanded output, so that the
good market is clear (condition 4).The condition 5 expresses the fact that the
…rm expects an e¤ect of its employment level on the bargaining outcome, that’s
why wi depends on ni .
In order to solve this problem, we consider the following equivalent variational problem:
Z 1
n_ i + sni
M ax
e rt f (ni ) c
dt;
(6)
ni
q( )
0
n_ i + sni
q( )
with ni (0) = ni0 and 0
vim , and where f (ni ) =
pi
(yi )yi (ni )
P
wi (ni )ni
For convenience, we rewrite the variational problem as follows:
Z 1
max
G(t; ni ) + H(t; ni )n_ i dt
n
(7)
0
cs
c
ni and H(t; ni ) = e rt
q( )
q( )
The Euler …rst order condition entails that the optimal solution of the problem of the …rm i; ni (t); is such that:
where G(t; ni ) = e
rt
f (ni )
@G(t; ni )
@H(t; ni )
=
@ni
@t
,
(s + r)c
c
@f (ni )
=
+
@ni
q( )
q( )
(8)
(9)
The expression (9) can be rewritten:
@ pPi (yi ) @yi (ni )
pi
@yi (ni )
yi (ni ) + (yi )
@yi
@ni
P
@ni
6 The
wi (ni )
@wi (ni )
(s + r)c
c
ni =
+
@ni
q( )
q( )
(10)
t index is removed for more convenient notations.
6
@ pPi (yi ) yi (ni )
1
=
; the previous equality becomes:
@yi pi (yi )
P
82
3
<
pi
@w
(n
)
c
c
i i
5
4wi (ni ) +
(yi ) =
ni + (r + s)
+
P
1:
@ni
q( )
q( )
As
@yi (ni )
@ni
The …rm …x its price by taking a mark-up equal to
9
1=
;
. The expression
1
between brackets represents the marginal cost of labor: wi (ni ) the unit cost of
c
@wi (ni )
labor, (r + s)
the search cost of an additional worker and
ni the
q( )
@ni
e¤ect on the wage bargaining outcome of an additional worker. This equation
can be also rewritten as an expression relating the …rm’s optimal choice of
employment and real wage:
wiD (ni ) =
@yi (ni )
1 pi
(yi )
P
@ni
@wiS (ni )
ni
@ni
(r + s)
c
q( )
c
q( )
(11)
This expression corresponds to the optimal "pseudo" labor demand of the
…rm i. In (11) and hereafter, we note wiD (ni ) the real wage of the "pseudo"
labor demand and wiS (ni ) the real wage of the "pseudo" labor supply, even
@wiS (ni )
though both of them result from the bargaining process. The derivative
@ni
shows us that the …rm expects that the bargaining result is in‡uenced by its
employment level. We also note through
c
that the labor demand is driven
q( )
by the evolution of the labor market thightness .
Now, we need to model the wage bargaining in order to obtain the "pseudo"
supply of labor wiS (ni ) and …nalize the determination of wiD (ni ).
1.3
Intra-…rm bargaining
The wage negotiation takes place between the worker and the …rm (individual
bargaining). Given the assumption of identical workers, the negotiated wage is
the same for all workers of the …rm.
The …rm opens vacancies which are matched to the pool of unemployed
workers and lead to employment. Given that it takes one period to …ll a vacancy,
the actual employment is …xed at the same time that the negotiation. That’s
@wiS (ni )
why we have the derivative
in (11). Thus, the …rm determines its
@ni
employment level heeding its e¤ect on the wage negotiation.
Let Ji and Vi the expected discounted value of pro…t from an additionnal
…lled job7 and a vacant one, and let Ei and Ui the expected discounted value of
7 We must keep in mind that we have large …rms here and not only one job for each …rm
as in Pissarides (2000).
7
income stream of an employed and unemployed worker. Consequently, during
the bargaining, the …rm and its worker share themselves the total surplus of the
matching Si = Ji + Ei Ui Vi .
The negotiated wage solves the following surplus Nash sharing rule:
Ei
Ui =
(Ji
1
Vi )
(12)
Where 2 [0; 1] represents the bargaining power of workers.
Since the …rms can open as many vacancies as necessary to obtain its optimal
employment level, the free entry condition drives the value of Vi to zero and
thus Ji , the expected marginal pro…t for the optimal employment level is equal
to the expected recruitment cost of workers8 :
Ji =
c
@V (ni0 )
=
@ni0
q( )
(13)
c
q( )
(14)
what entails:
Ji =
According to the …rst order conditions of the …rm’s program, we have also:
2
3
S
1
p
@y
@w
(n
)
c
i
i
i
i
5 =(r + s)
Ji = 4
(yi )
(15)
wiS (ni )
ni +
P
@ni
@ni
q( )
The expected discounted utility of a job for a worker Ei satis…es the Bellman
equation:
rEi = wiS (ni )
s [Ei
Ui ] + E i
(16)
The expected discounted utility of an unemployed worker Ui satis…es also
the following Bellman equation:
rUi = b + q( ) [Ei
Ui ] + U i
(17)
where b represents the unemployment bene…ts. After substitution of (12)
and (13) in (17), we can rewrite the expression rUi as follows:
rUi = b +
c + Ui
1
(19)
After di¤erentiation of (12)9 we can also write:
Ei
Ui =
1
Ji =
8 See
1
c
q( )
(20)
Appendix B on the …rm problem for futher details.
that the negotiated wage is a jump variable, this sharing rule holds also in rates of
change (see Pissarides , 2000).
9 Given
8
The expressions (12), (16), (19) and (20) lead to the following …rst order
di¤erential equation:
wiS (ni ) = (1
@wiS (ni )
ni + c
@ni
@yi
1 pi
(ni )
P
@ni
)b +
(21)
Its solution10 gives us the expression of the wage resulting from the wage
negotiation:
wiS (ni ) = (1
)b+
@yi
1 pi
(yi )
+ c ; with
P
@ni
=
+
(22)
The negotiated wage is independent from any dynamic of theta. Thus, we
…nd the Pissarides’result which asserts that the negotiated wage holds both in
and out of steady state11 . This expression represents the "pseudo" labor supply
at the …rm level.
Now, we compute the derivative of the equation (22) to insert it in the
equation (11):
1h
@wiS (ni )
=
@ni
(
ip
1)
i
P
(ni )
@yi
n
@ni i
1
(23)
This equation represents the hiring externality due to the intra…rm bargaining wage setting as well as the slope of the individual wage curve (22)12 . The
substitution of (23) in (11) gives us the "pseudo" labor demand at the …rm level
according to the wage negotiation result:
wiD (ni ) =
1.4
1 pi
@yi
(yi )
P
@ni
(r + s)
c
c
+
q( )
q( )
(24)
General equilibrium and steady state
At the general symmetric equilibrium all the …rms and workers are identical.
Thus, given the assumption on the labor force size and …rms distribution, we
pi
have
= 1; yi = Y , ni = n, vi = v; Ji = J; Ei = E, Ui = U and u = 1 n.
P
Thus, we obtain the following general equilibrium equalities:
1 @y
@n
wD (n) =
wS (n) = (1
With
=
(r + s)
)b +
c
c
+
q( )
q( )
(25)
1 @y
+ c
@n
(26)
+ :
1 0 See
appendix C for the detailed resolution.
the wage curve is independent of the law of motion of ni and :
1 2 See appendix D for an analysis of the sign of (23).
1 1 Indeed,
9
Given that wD (n) = wS (n), we can deduce the law of motion of the labor
market tightness at the general equilibrium and, thus, obtain the following
dynamic system:
8
q( )
c
1 @y
<
=
(1
) b + c + (r + s)
(1
)
c
q( )
@n
(27)
:
n = q( )v sn
At the steady state, the employment and labor market tightness are constant,
so n = = 0. Hence, the general equilibrium equalities at the steady state
becomes:
c
1 @y
(r + s)
(28)
wD (n) =
@n
q( )
wS (n) = (1
1 @y
+ c
@n
)b +
(29)
Thus, we obtain the same equations than those found in Cahuc and Wasmer
(2004).
v
v
Since u = 1 n, the tightness of labor market can be written: = =
:
u
1 n
And given the constancy of employment rate at the steady state13 , we have:
n = q( )v
sn = 0 , v =
sn
q( )
(30)
As q( ) =
, the expression (30) can be written : v = sn .
Using the previous expressions of and v, we obtain the following expression
of and q( ) as functions of n only:
(n) =
q( (n)) =
1=1
sn
1
(31)
n
=1
sn
1
n
(32)
@q( (n))
@ (n)
With
< 0 and
> 0: Thanks to the equations (31) and (32),
@n
@n
we can also rewrite the equilibrium equalities as functions of employment rate
only:
=1
1
sn
wD (n) =
A n 1 (r + s)c
(33)
1 n
"
#
1=1
1
sn
wS (n) = (1
)b +
A n 1+c
(34)
1 n
with
=
+ :
1 3 This corresponds to the ‡ow equilibrium condition which implies a pseudo Beveridge
Curve that is to say a positive relation between the vacancies and employment.
10
2
Equilibrium
In this section, we study the possible equilibrium cases which can occur according to the nature of returns to scale. However, before doing this, we must de…ne
clearly what equilibrium means here. In our model, an equilibrium corresponds
to any couple of variable (n; w) at the steady state for which both …rms and
workers behave optimally. As a consequence, it corresponds to the intersections between the curves represented by the equation (33), the product market
equilibirum locus, and the equation (34),the labor market equilibrium locus.
2.1
Constant or decreasing returns to scale and unique
equilibrium
Proposition 1 When the returns to scale are decreasing or constant (0 <
1), there is an unique equilibrium on the interval n 2 ]0; 1[.
In the space (n; w), the expression wD (n) is decreasing when 0 <
Indeed, we have:
@wD (n)
@n
1
=
1
=
A (
1)n
A (
1)n
2
(r + s)c
2
(r + s)c
"
@q( (n))
q( (n))
@n
2
s
1
(1
sn
n)2
1
6
6 1:
(35)
2
1#
1
n
@wD (n)
1
is always negative when 0 < 6 1 because
A (
@n
@q( (n))
1)n 2 is always negative and
q( (n)) 2 is always positive. On the
@n
interval n 2 ]0; 1[, the expression wS (n) is U shaped when 0 <
< 1 and
increasing when = 1. We can conclude it according its limits’computations:
Consequently,
lim wS (n)
=
+1
lim wS (n)
=
(1
n!0
n!0
and
)b +
lim wS (n) = +1;
n!1
1
when 0 <
<1
A and lim wS (n) = +1; when
n!1
=1
Now, to show the existence of an unique equilibrium and so of one intersection between the curves in the space (n; w), we must demonstrate that the
curve wD (n) is above the curve wS (n) when n tends toward zero.
In the case where 0 < < 1, both expressions tend to +1 when n tends to
1
zero: This results from the following part of the expressions:
A n 1 for
wD (n) and
1
1
A n
for wS (n), but given that
wD (n) > wS (n) when n tends to zero.
if and only if b <
1
= 1, wD (0) =
In the case where
1
A.
11
2 [0; 1] we have always
A > wS (0) = (1
)b +
1
A
2.2
Increasing returns to scale and multiple equilibria
Proposition 2 When the returns to scale are increasing (
either multiple distinct equilibria or none.
> 1), there are
To show the existence of multiple equilibria, we need to show the multiplicity
of intersections between the curves wD (n) and wS (n) in the space (n; w):
The curve wS (n) is strictly increasing in the space (n; w) when the returns
to scale are increasing. Indeed, its derivatives gives:
@wS (n)
@n
1
=
=
1
A (
A (
2
1)n
1)n
2
+c
+ c
@ (n)
@n
(36)
s
1
(1
sn
n)2
1
1
n
@ (n)
> 0, (33) is always positive when > 1.
@n
D
The curve w (n) is increasing and, then, decreasing when n becomes high
(near to one). We can note this through its partial derivatives (35) which becomes negative at a certain treshold of n named
n14 . Truly, when n > n, #
"
2
1
1
1
s
sn
A ( 1)n 2 becomes inferior to (r+s)c
1
(1 n)2 1 n
and (35) becomes negative.
@wD (n)
Thus, if the slope of wD (n) is steeper than wS (n) for n < n , i.e.
>
@n
S
@w (n)
when n < n, and if wD (0) < wS (0), the curves must intersect at
@n
least twice in the space (n; w). For increasing returns to scale ( > 1), we
have always: wD (0) = 0 < wS (0) = (1
)b. However, when the returns to
@wS (n)
@wD (n)
>
is not veri…ed15 . As a
scale become to high, the inequality
@n
@n
consequence, when the returns to scale are too increasing, the economy exhibits
no equilibrium, otherwise it exhibits at least two.
Given that
3
Numerical simulations
In this section, we run numerical simulations in order to give support to the
previous propositions and to assess the e¤ects of:
the size of returns to scale on the equilibrium possibilities
the competition degree on the product market
1 4 This feature will be illustrated in the numeric simulations. The treshold for which (35)
becomes negative depend on the value of various parameters.
1 5 This statement will be demonstrated in the section of numerical simulations.
12
an unemployment bene…ts increase
an increase of workers’bargaining power
on labor market perfomances.
The model period is one year. We don’t calibrate the model according to
empirical studies on a particular economy, but our parametrization choices are
based on previous works (Ebell and Haefke, 2003) and corresponds to realistic
one. However, some di¢ culties appear regarding some parameters. We will
discuss this issue below.
In all following graphs, the black curves represent the equation wD (n) and
the grey curves the equation wS (n):
3.1
Equilibrium
In this part, we …x a set of parameters and we move the size of returns to scale
in order to veri…ed the propositions 1 and 2. The parametrization is given in
the following table:
Table 1: Parameters values
A=1
= 0:5
= 0:5
s = 0:1
c = 0:2
r = 0:04
b = 0:3
Technological level (normalized)
Workers bargaining power (standard)
Elasticity of matching function
Job destruction rate
Vacancy cost
Discounted rate
Unemployment bene…t
For more simplicity, we normalized to one A, the parameter of technological
level. Furthermore, we set = = 0:5 and, thus, we use standard values and
impose Hosios condition (1990) for constant returns to scale case16 . The job
desctruction rate value corresponds to a destruction of 10% of existing jobs each
year and the discounted rate one to a 4% annual interest rate. The parameter
b is commonly interpreted as the monetary compensation for the unemployed.
Usually, it’s more easy to consider the replacement rate, i.e. the ratio of bene…ts
to wage, during calibration. According OECD reports, the average replacement
rate can be very di¤erent from a country to another and from a family type to
another but in average it can be included between 20% and 80%. In our model,
the wage is determined endogenously and so when we …x a value for b we can give
the replacement rate value after its determination. Here we …x the parameter
1 6 Hosios (1999) identi…ed a general condition underwhich all the externalities of search
process are internalised and all decisions are e¢ cient: the matching elasticity with respect
to unemployment must be equal to the worker’s share of the match surplus in the case of
constant returns to scale in the production technology.
13
b to 0.3. The same problem appears for the parameter of vacancy cost c, we
…x its value at 0.2 here. The parameter > 1, which represents the demand
elasticity of the good supplied by the …rms, translates the degree of competition
on the good market in our economy17 . Indeed, when the competition is strongly
monopolistic it tends to 1 and, inversely, when the degree of competition comes
near to perfect competition, it tends to +1. Through the expression "degree
of competion" we means the degree of product market competition, that is to
say the power that the …rms have on the good prices. In this …rst subsection,
this parameter is …xed to 2 (monopolistic competition). Later, we will increase
its value in order to check the e¤ects of the good market competition.
3.1.1
Decreasing and constant returns to scale
In this part, we will demonstrate graphically thanks to numerical simulations
that we have always an unique equilibrium in the case where the returns to scale
are decreasing or constant.
w
1.0
0.8
0.6
0.4
0.2
0.0
0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98
n
Figure 1: Decreasing returns to scale
w
= 0:5
0.8
0.6
0.4
0.2
0.0
0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98
n
Figure 2: Constant returns to scale
1 7 This
=1
feature comes from the assumption on …rms distribution on the interval [0; 1] :
14
The …gures 1 & 2 con…rm the algebra proof of unique equilibrium of the
previous section and give support to the proposition 1. We …nd the usual results
that the strategic complementarity introduced by the monopolistic competition
isn’t strong enough to involve the presence of multiple equilibria. The graphs
show also that the equilibrium in the case of decreasing returns to scale is worse
than the one in the case of constant returns to scale. Indeed, the employment
rate and the wage are higher in the second one than in the …rst one.
According to our parameterization and equilibrium outcomes, we compute
the reservation wage of workers U 18 to compare our …ndings with those of Stole
and Zwiebel (1996), and Pissarides (2000). In the case of decreasing returns, we
…nd a mix of them. Indeed, as in Stole and Zwiebel (1996) where returns to scale
are decreasing, we note the absence of rents for workers19 . Due to the presence
of hiring externality which depress wages when an additional worker is hired20 ,
the …rms exploit the decreasing returns and overemploy in order to moderate the
workers’wages. However, we …nd that equilibrium wages is lower than marginal
labor productivity in both two cases due to the presence of hiring cost. Thus,
our …ndings are similar to those of Pissarides (2000) on this point. In the case of
constant returns, our results are also consistent with those of Pissarides (2000)
concerning workers’ rents. In spite of intra-…rm bargaining, workers get rents
due to the constancy of returns ruling out strategic employment level which
depress wages until reservation wage21 .
3.1.2
Increasing returns to scale
Here, we run several numerical simulations with di¤erent values of > 1: By
this way, we give support to the presence of multiple equilibria and to absence
of them (proposition 2). These simulations also allow to investigate the e¤ects
of the size of returns to scale on the equilibrium outcomes.
1 8 At the each equilibrium, the reservation wage can be deduced from the equilibrium values
of w and n :
1 9 The equilibrium wage is equal to the reservation wage in the case of decreasing returns
what translates an absence of rents given that the workers earn what they would be expected
earn if they were unemployed.
@wiS
2 0 We show easily that
< 0 when
1, so hiring an additional worker will decrease
@ni
the wage for all workers.
@wiS
@wiS
2 1 We have
>
: As a consequence, even if the …rms overemploy, wages
@ni
@ni
<1
=1
cannot diminish until the reservation wage.
15
w
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Figure 3: Increasing returns to scale
w
0.8
0.9
n
= 1:5
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Figure 4 : Increasing returns to scale
0.8
0.9
n
=2
w 1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Figure 5 : Increasing returns to scale
0.8
0.9
n
=3
The previous graphs show the existence of two positive equilibria. These can
be characterized as follow: one as the "low equilibirum", where both employ-
16
ment rate and wage are low, and another as the "high equilibrium", where both
employment rate and wage are high.
Concerning the workers’rents, we …nd Stole and Zwiebel’s (1996) result at
the low equilibrium and the Pissarides’(2000) one at the high equilibrium. At
both equilibria, the marginal productivity of labor is always greater than the
equilibrium wage. At the high equilibrium, the workers get rents which are
increasing with the size of returns. Indeed, we observe a higher wage when the
returns’size increases, whereas the equilibrium employment remains unchanged.
Nonetheless, at the low equilibrium, the …rms behave strategically and exploit
increasing returns to scale of labor by underemploying and, as a consequence,
workers get no rents. At this equilibrium, an increase of returns to scale have
just an e¤ect on employment, it increases, but the wage remains unchanged and
equal to reservation wage.
w
2.5
2.0
1.5
1.0
0.5
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Figure 6 : Increasing returns to scale
0.8
0.9
n
=4
In the case where the returns to scale are too high we have no equilibrium.
This case gives additional support to the proposition 2.
In what follows, due to the complexity of equilibrium equalities, we run
numerical simulations with the intention of managing a comparative statics
analysis.
3.2
Higher degree of competition
In this section, we investigate the e¤ects of a higher competition degree on
the good market on the labor market outcomes. In order to do that, we use
the parameters values given in the table 1 as well as two disctinct values of
translating the case of monopolistic competition and of more competitive good
market. The solid lines represent the case where = 2 (the benchmark case),
that is to say when the competition is "very" monopolistic and the dashed lines
represent the case where the degree of competition is stronger, = 100:
In their paper, Ebell and Haefke (2003) identify two channels by which
competition a¤ects employment22 in a model with intra…rm bargaining. The
2 2 They
study the e¤ect of competition on unemployment; but, given our assumption on the
17
…rst one is the output expansion channel: a stronger competition involves a
higher pro…t-maximizing output and so a higher employment level. The second
one is the hiring externality channel. We have seen in the previous subsection
how the …rms behave strategically in presence of intra…rm bargaining due to
the presence of hiring externality. This second channel can either reinforce or
diminish the …rst channel e¤ect on employment23 .
3.2.1
Decreasing returns to scale
In the case of decreasing returns to scale as well as in the case of constant
returns to scale24 , a higher competition on the product market improves both
employment rate and real wage. These results are quite similar to those of
Blanchard and Giavazzi (2003) which state that stronger competition on good
market leads to higher real wages and lower equilibrium unemployment rates.
The …gure 7 illustrates the case of decreasing returns to scale.
w
1.0
0.8
0.6
0.4
0.2
0.0
0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98
n
Figure 7: Stronger competition when
= 0:5
A stronger competition on the product market decreases the market power on
price of each …rm. As a result, at the general equilibrium, the general price level
is lower and the real wages increases. Then, involving a higher pro…t-maximizing
output, stronger competition leads to higher employment level (…rst channel).
Furthermore, we show easily that the hiring externality is less important when
the degree of competition increases25 in this case. Given that the employment
rate increases here, we conclude that the …rst channel is stronger than the second
one here.
labor force, we can compare their …ndings with ours.
2 3 This depends on the returns to scale features.
2 4 The case of constant returns to scale is not illustrated here, but the results are the same
than in the case of decreasing returns, that
" is to say
# a higher wage and a higher employment.
@wiS (ni )
@
@ni
2 5 We show in the appendix E that
< 0 when 6 1:
@
18
3.2.2
Increasing returns to scale
In the case where the returns to scale are increasing, we …nd the same results
than Blanchard and Giavazzi (2003) at the high equilibrium only. Indeed, according to the …gures below, we note that with a stronger competition on the
good market the high equilibrium displays a higher wage rate than in the benchmark case and an unchanged employment rate, whereas the low equilibrium displays a lower employment rate than in the benchmark case and an unchanged
wage.
w
2.0
1.5
1.0
0.5
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Figure 8: Stronger competition when
w
0.8
0.9
n
= 1:5
2.0
1.5
1.0
0.5
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Figure 9: Stronger competition when
3.3
0.8
0.9
n
=3
Unemployment bene…ts increase
Here, we investigate the e¤ects of an increase in unemployment bene…ts. We
still use the parameters values of table 1 and
= 2 in the benchmark case
(solid curves) and we compare it with the case where unemployment bene…ts
are higher, b = 0:4 (dashed curves). As previously, we …rst check the case of
decreasing returns and, then, the case of increasing returns.
19
3.3.1
Decreasing returns to scale
w
1.0
0.8
0.6
0.4
0.2
0.0
0.5
0.6
0.7
0.8
Figure 10: Higher allocations
0.9
1.0
n
= 0:5
The …gure 11 shows that raising unemployment bene…ts involves a reduction in the employment rate and an increase in real wage. Actually, higher
unemployment bene…ts imply higher reservation wage for workers and reduce
incentives to work. As a result, the new wage is higher and always equal to the
reservation wage here; while the employment rate is lower. We note the same
widespread outcome when the returns to scale are constant.
3.3.2
Increasing returns to scale
In the case of increasing returns to scale, we don’t …nd the previous result. The
…gures 12 & 13 show us that an increase in unemployment bene…ts has:
- a tiny negative e¤ect on employment at the high equilibrium
- a positive e¤ect both on the employment rate and on the wage at the
low equilibrium.
w
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Figure 11: Higher allocations
20
0.7
0.8
= 1:5
0.9
n
w 1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Figure 12 : Higher allocations
0.8
0.9
n
=3
We have found in the subsection 3.1 that workers get rent at the high equilibrium which is increasing with the returns’size. The extent of this rent implies
a non signi…cant e¤ect of a moderate increase in unemployment bene…ts. Indeed, the rent is so high that higher unemployment bene…ts have a tiny e¤ect
on incentives to work and, the greater the returns’size is, the tinier this e¤ect
is. Only an unemployment bene…ts increase superior to the workers’ rent can
have an e¤ect at the high equilibrium26 .
3.4
Stronger bargaining power of workers
Here, we investigate the e¤ects of an increase in the workers’bargaining power
. We still use the parameters values of table 1 and = 2 for the benchmark
case (the solid curves) and, next, we raise the parameter at the value 0:8 to
translate a stronger bargaining power of workers27 (the dashed curves).
2 6 In this case, the e¤ect would be the same that the common result, that is to say a decrease
in employment rate and a wage increase.
2 7 Note that now the Hosio (1990) condition do not hold.
21
3.4.1
Decreasing returns to scale
1.0
0.8
0.6
0.4
0.2
0.0
0.80
0.82
0.84
0.86
0.88
0.90
0.92
0.94
Figure 13: Stronger bargaining power
0.96
0.98
n
= 0:5
In this case, a stronger bargaining power of workers involves a higher wage
and a lower employment rate and implies the same results than those found in
the literature.
3.4.2
Increasing returns to scale
Here, whatever the equilibrium, the e¤ect of a stronger bargaining power on the
wage goes in the same direction, whereas it is opposite on the employment rate.
w
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Figure 14: Stronger bargaining power
22
0.8
0.9
= 1:5
n
w 1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Figure 15 : Stronger bargaining power
0.9
n
=3
Indeed, according the …gures 15 & 16, we note that at the high and low
equilibrium a stronger bargaining power implies a lower wage28 . Inversely, it
implies an higher employment rate at the low equilibrium and a lower one at
the high equilibrium.
4
Stability analysis and discussion
4.1
Stability analysis
Running a stability analysis of the model at the general equilibrium implies the
speci…cation of the dynamics of and n. These are given in the system (27):
8
q( )
1 @y
c
<
=
(1
)
(1
) b + c + (r + s)
c
q( )
@n
:
n = q( )v sn
According to the propositions 1 & 2 and the previous numerical simulations,
we note that there exists an unique equilibrium when the returns to scale are
decreasing or constant and two disctinct equilibria, a low equilibrium and a high
equilibrium, when the returns to scale are increasing (but not too). On the basis
of the parameters’values used in the previous section, we can …nd numerically
that:
If
1; the unique interior equilibrium is a saddle point.
If > 1; the low interior equilibrium is unstable and the high one is a
saddle point29 .
2 8 At
the low equilibrium, its negative e¤ect on wage is tiny.
saddle point arises in both cases because n, the employment rate, is a sticky and
stable variable; whereas v, vacancies, is a forward-looking and unstable variable (Pissarides,
2000).
2 9 The
23
For all , there exists two corner equilibria which are stable and for which
n = 0 or 1:
The equilibrium paths of the variables and n are plotted (arrows) in the
…gures 16 & 17 in case of decreasing and increasing returns to scale.
Figure 16: Equilibrium paths when
24
= 0:5
Figure 17: Equilibrium paths when
= 1:5
We can see on the two …gures that there exists only one equilibrium path
(or saddle path) for which n converges to the saddle point in both cases. As in
Pissarides (2000), this saddle path corresponds to the curve = 0: According
to the …rst order condition of the …rm’s problem, the optimal employment of
the …rm i; as well as the initial state of its employment level ni0 ; is always
on the -stationary (i.e. the curve = 0) in the absence of expected change
in the exogenous variable 30 . Thus, at our general symmetric equilibrium,
the economy is always on the saddle path and converges to the saddle point
through variations of employment only (and open vacancies). We focus on this
case here31 .
4.2
Discussion
In this part, we discuss the comparative statics results. Since the high equilibrium is the one where the economy converges in the case of multiple equilibria
with perfect foresight, we focus the discussion on this equilibrium and rule out
the others. In the case of unique equilibrium, we ruled out also the corner equilibria and consider only the interior one Thus, we compare both the sign and
the extent of the e¤ects on employment and wage of the three economic policies
3 0 We
must keep in mind that the variable is exogenous at the …rm level.
case where the …rms expect any change in is worth to be investigated, but it’s not
the purpose of the paper.
3 1 The
25
reported previously in the case of unique and multiple (interior) equilibria. The
tables 2 & 3 summarize our …ndings.
Table 2: E¤ects’sign
61
Product market deregulation:
>0
Unemployment bene…ts increase:
b>0
Workers’bargaining power increase:
>1
@n
@w
>0
>0
@
@
@n
@w
>0
>0
@
@
@w
@n
<0
>0
@b
@b
@n
@w
60
'0
@b
@b
@n
@w
<0
>0
@
@
@n
@w
<0
<0
@
@
>0
Table 3: Comparison of the e¤ects’extent
Employment Rate
>0
b>0
>0
@n
@
@n
@b
@n
@
>
61
>
61
<
61
Real Wage
@n
@
>1
@n
@b
>1
@n
@
>1
@w
@
>
61
@w
@
>1
When the e¤ects of a economic policy go in the same direction in both
case, they are always higher in the case of unique equilibrium than in the case
of multiple equilibria except for an increase in the workers’ bargaining power
(table 3). Indeed, we note that higher returns to scale entail lower impact of
some economic policies.
We note also that in the case of multiple equilibria an increase of unemployment bene…ts has no e¤ect on the real wage and a tiny one on employment
whereas it entails a decrease in employment and an increase in real wage in
the case of unique equilibrium according to our calibration. This feature comes
from the size of workers rents. Indeed, an increase of unemployment bene…ts
will has as much more impact as it will reduce the workers wage rents.
Conclusion
In this paper, we have used a model similar to this of Ebell and Haefke (2003)
with monopolistic competition on the product market, matching frictions and
26
intra-…rm bargaining on the labor market. The main contribution of our paper
is to allow for increasing returns to scale in production technologies and to investigate in a such case the e¤ects of deregulation on product market, increase
of unemployment bene…ts and stronger bargaining power of workers on labor
market performances in a such framework. To do that, we have also specify the
out of steady state dynamics of the labor market tightness as in the Mortensen’s
paper. Our …ndings have shown that some results of Stole and Zwiebel don’t
hold in the case of multiple equilibria (absence of rent for workers due to the
intra-…rm bargaining). We have also demonstrated that the impact of some
economic policy are lower in the case of multiple equilibria than in the case
of unique equilibrium. Future researches should more investigate how the expectations of agents on the exogenous variable as in‡uence the dynamics of
employment and unemployment in this kind of economy.
References
[1] Blanchard, O. and Giavazzi, F. (2003), "Macroeconomic E¤ects of Regulation and Deregulation in Goods and Labor Markets", Quarterly Journal
of Economics 118, 879-907;
[2] Cahuc, P. and Wasmer, E. (2001a), "Labor Market e¢ ciency, Wages and
Employment when Search Frictions Interact with Intra…rm Bargaining",
IZA Discussion Paper N 304, June 2001;
[3] Cahuc, P. and Wasmer, E. (2001b), "Does intra-…rm bargaining matter in
the large …rm’s matching model?", Macroeconomic Dynamics, 5, 742-747;
[4] Cahuc, P. and Wasmer, E. (2004), "A theory of wages and labour demand
with intra-…rm bargaining and matching frictions", CEPR Discussion Paper N 4605, September 2004;
[5] Clark, C.W. (1990), Mathematical Bioeconomics: The Optimal Management of Renewable Resources, John Wiley and Sons, New York;
[6] Cooper, R.W (1999), Coordination games: Complementarities and Macroeconomics, Cambridge University Press;
[7] Ebell, M. and Haefke, C. (2003), "Product Market Deregulation and Labor
Market Outcomes", IZA Discussion Paper N 957, December 2003;
[8] Hart, R.F. and Feightinger, G. (1987), A New Su¢ cient Condition for Most
Rapid Approach Paths, Journal of Optimization Theory and Applications,
Vol.54, 403-411;
[9] Pissarides, C.A. (2000), Equilibrium Unemployment Theory, Second Edition, Cambridge, The MIT Press;
[10] Mortensen, D.T. (1999), "Equilibrium Unemployment Dynamics", International Economic Review, 40(4), November 1999, 889-914;
27
[11] Stole, L. A. and Zwiebel, J. (1996), "Intra…rm Bargaining under Nonbinding Contracts", Review of Economic Studies, March1996, 63, 375-410;
Appendix
Appendix A: Determination of the demand for the …rm i’s
output
Households are both consumers and workers. They are risk neutral and have
Dixit-Stiglitz preferences over a continuum of i di¤erenciated goods uniformally
distributed on the interval [0; 1]. A representative household derives its demand
in good i by solving:
Z 1
1
1
(A1)
M ax
ci di
ci
0
R 1 pi
under the budget constraint 0 ci di = I.
P
Where > 1 represents the elasticity of substitution between goods, pi the
price of good i , P the price index and I the real income of the representative
household.
Given the absence of saving and the fact that the population of indentical
households is normalized to one, we obtain the aggregate demand for the good
i given as:
pi
yid = ci =
I
(A2)
P
Furthermore, assuming a clear product market, we have P I = P Y where I
represents the aggregate real income here and Y the aggregate output. Thus,
we can write (A2) as following:
pi
P
yid =
with the price index P =
Z
0
1
1
pi
di
Y
1
(A3)
:
The expression (A3) represents a standard monopolisitic competition demand function with an elasticity of subsitution among di¤erenciated goods given
by
:
Appendix B: Existence and uniqueness of the …rm’s problem
The …rm solves:
Z 1
max
e rt f (ni )
ni
0
c
n_ i + sni
q( )
dt;
ni (0) = ni0 ;
0
n_ i + sni
q( )
vim :
(B1)
28
The turnpike solution is given by ni (t):
ni (t) such that f 0 (ni ) =
(s + r)c
c_
+
q( )
q( )
(B2)
The following theorem called the most rapid approach theorem or turnpike
theorem (Clark,1990 and, Hart and Feightinger, 1987) gives the optimal solution
for this problem:
Theorem 3 If
Gn (t; n) > (<)Ht (t; n);
when
n > (<)n (t);
where ni (t) is the unique solution of (B2) and if for all admissible paths ni (t),
the following condition holds:
Z n (t)
c
(n (t) ni (t)) 0;
lim e rt
H(t; x)dx = lim e rt
t!1
t!1
q(
) i
n(t)
then, the optimal solution of problem (B1) is
i.e.:
8
>
0
si
>
>
>
>
>
>
>
>
>
si
<vim
vi (t) =
>
>
>
> n_i + sni
>
>
si
>
> q( )
>
>
:
the most rapid approach to ni (t)
ni > ni (t)
ni < ni (t)
(B3)
ni = ni (t):
We are going to verify if the hypothesis of this theorem holds with the wage
function wiS (n) given by (22). To begin with, let us compute
Ji =
@V (ni0 )
@ni0
which is used in the wage negotiation.
Suppose ni0 > ni (0), in this case the optimal control is zero and the optimal
path ni (t) is the solution of:
n_ i =
Let
nopt
i (t)
= ni0 e
st
sni ; ni (0) = ni0
this solution and
tm such that ni (tm) = nopt
i (tm):
(B4)
where tm is the time at which the optimal path meets ni (t) and note that
tm = tm(ni0 ). Consequently, we have
!
Z tm
Z 1
n_i (t) + sni (t)
opt
rt
rt
dt
V (ni0 ) =
e f (ni (t))dt +
e
f (ni (t)) c
q( )
0
tm
(B5)
29
and di¤erentiating with respect to ni0 , yields
Z tm
@V (ni0 )
@f (nopt
i (t))
e rt
=
dt + e
@ni0
@n
i0
0
Z
@ f (ni (t))
c
rtm
f (nopt
i (tm))
@tm
+
@ni0
n_i (t)+sni (t)
q( )
n_ (tm) + sni (tm)
e rt
dt e rtm f (ni (tm)) c i
@n0
q(tm)
tm
(B6)
s tm
(tm)
=
n
e
and
by
(B4):
As ni does not depend on ni0 , ni (tm) = nopt
i0
i
1
@tm
n_i (tm)
=e
@n0
we obtain that
rtm
rtm @tm
ni0 se
e rtm
@tm
=
@ni0
n_i (tm) + ni0 se
@n0
rtm
;
;
so that (B6) becomes:
@V (ni0 )
=
@ni0
Z
tm
e
(r+s)t 0opt
f
(t)) dt + e
(r+s)tm
0
c
:
q( )
(B7)
When tm goes to zero with ni0 > ni (0), we have:
+
@V (ni (0) )
c
:
=
@ni0
q( )
(B8)
The same argument is valid when ni0 < ni (0), and
@V (ni (0) )
c
:
=
@ni0
q( )
Remark 4 Note that for this singular control problem, (but only when ni0 =
c
ni (0)) or, in other words, when starting on ni (t))
is the value of the coq( )
state variable of the control problem only when ni0 = ni (0) or, in other words,
when starting on ni (t).
H(ni ; vi ; ; ) = (f (ni )
cvi ) + (q( )vi
sni );
actually:
@H(ni ; vi ; ; )
=0
vi
()
=
c
:
q( )
Now, we verify that the equation (B2) (or equation (10) in the paper’s body)
has an unique solution when the wage is given by the negotiated wage (equation
(22)). The substitution of (22) in (B2) gives:
n
+
1
=
(1
)b +
(1
c + c(r + s)=q( )
c _ =q( )
= K:
1
1=
1=
)(
1)A
Y
(B9)
As a result, there exists an unique positive solution to this equation when
K > 0.Sine
> 1;this statement always holds when c _ < (1
)bq( ) +
q( )c + c(r + s).
30
!
@tm
:
@n0
Appendix C: Solving the di¤erential equation
The di¤erential equation to be solved is:
wi (ni ) = (1
)b +
yi
1 pi
(ni )
P
ni
c+
wi (ni )
ni
ni
(C1)
The method of resolution is standard and follows Cahuc and Wasmer (2004).
Initially, we can disregard the term which don’t depend on ni (the constant
pi
yi
1
1
1
Y ; the
term) and add it back in later. Given that (yi )
= A 1 ni
P
ni
equation (C1) becomes:
1
wi (ni ) =
1
A1
ni
1
wi (ni )
ni
ni
1
Y
(C2)
And can be rewritten as follows:
wi (ni )
wi (ni )
+
ni
ni
1
1
A1
ni
2
Y
1
=0
(C3)
The homogenous version of (C3) is:
wi (ni )
wi (ni )
+
=0
ni
ni
which has the solution:
(C4)
1
wi (ni ) = Kni
(C5)
We take the derivative of (C5), using the fact that K may depend upon ni :
wi (ni )
=
ni
1
K ni
1
1
K
ni
1
+ ni
(C6)
Now, we substitute (C5) and (C6) in (C3) and we obtain:
1
ni
,
Given that
(B7) gives:
K
ni
1
A1
1
K
=
ni
1
A1
1
pi
yi
(yi )
= A1
P
ni
1
ni
1
2
ni
2
ni
Y
1
+1
=0
Y
1
1 pi
yi 1
(yi )
n +J
P
ni i
where J is a constant of integration and =
substitute (C8) in (C5) and we obtain:
yi
1 pi
(yi )
+ Jni
P
ni
31
(C7)
Y , the integral over both sides of
K=
wi (ni ) =
1
(C8)
+ . Now, we
1
(C9)
Following Cahuc and Wasmer (2004), the terminal condition lim ni wi = 0,
ni !0
which reports the fact that the …rm-level bargained wage should not explode as
…rm-level employment ni approaches zero, implies that J = 0. As a consequence,
the constant of integration can be withdrawn and ,after adding back the constant
term, we obtain the following solution for the di¤erential equation (C1):
wi (ni ) = (1
with
=
)b +
1 pi
yi
(yi )
P
ni
c+
(C10)
+ :
Appendix D: Hiring externality analysis
The sign of the hiring externality given by the expression () depends on the sign
of the following expression:
a( ; ) = (
According the values of
or null. Indeed, we have:
- a( ; ) = 0 if
=
- a( ; ) > 0 if
>
- a( ; ) < 0 if
<
(D1)
1)
and , the function a( ; ) is positive, negative
1
1
1
And graphically, we have:
α
a>0
a=0
1
a<0
σ
1
Figure D1: Sign of a( ; )
32
Thus, when the size of returns to scale is inferior (superior) to the price
mark-up of the …rm, the hiring externality is negative (positive).
We can also conclude that whatever the degree of competition on the product
market, the hiring externality is always negative when the returns to scale are
decreasing or constant.
Appendix E: Hiring externality and competition degree
We investigate here how the hiring externality varies when the competition
degree increase. To do that, we need to compute the following derivative:
@
@wiS (ni )
@ni
@
1h
@
=
(
ip
1)
i
P
@
(ni )
@yi
n
@ni i
1
(E1)
Let:
1
u
=
v
=
(
w
=
pi
@yi
(ni )
n
P
@ni i
1)
1
The derivative of (1) is given by u0 vw + uv 0 w + uvw0 :
Given that:
u0
=
v0
=
w0
=
+(
1)(
+ 1)
2
> 0; since
>0
hp
ip
1
@yi
i
i
ln
(n
)
(ni )
n
i
2
P
P
@ni i
2 [0; 1]
2
1
= 0, since
pi
(ni ) = 1 due to the symmetrical …rms assumption
P
The derivatives of (E1) is given by this following expression after arrangements:
@
@wiS (ni )
@ni
@
=
+(
1)(
(
1)
+ 1)
@wiS (ni )
(
+
@ni
1)
2
pi
@yi
(ni )
n
P
@ni i
Thus, (E1) is negative when the hiring externality is negative, that is to say
@wiS (ni )
< 0 and positive otherwise.
@ni
When the returns to scale are decreasing or constant ( 6 1), the hiring
externality is always negative32 and the expression (E1) too.
3 2 See
appendix D.
33
1